Nordhaus-Gaddum problem in term of $G$-free coloring

Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi : V(H) \longrightarrow {1,2,\ldots, k}$,ifeachcolorclassinducesa$K_2$-freesubgraph\mathrm{.}Foragraph$G$oforderatleast$2$,a$G$-free$k$-coloringof$H$,isamapping$\pi : V(H) \longrightarrow {1,2,\ldots,k}$,sothattheinducedsubgraphbyeachcolorclassof$\pi$, contains no copy of $G$. The $G$-free chromatic number of $H$, is the minimum number $k$, so that it has a $G$-free $k$-coloring, and denoted by ${\chi }_G(H)$. In this paper, we give some bounds and attributes on the $G$-free chromatic number of graphs, in terms of the number of vertices, maximum degree, minimum degree, and chromatic number. Our main results are the Nordhaus-Gaddum-type theorem for the $\text{G}$-free chromatic number of a graph.